Discrete Convolution-fast Fourier Transform Research Articles

Lubrication Prediction of Sphere‐Gradient Coated Half Space Interfaces

ABSTRACTFunctionally graded coating (FGC) has played a pivotal role in numerous engineering applications owing to its exceptional properties. This work proposes a novel elastohydrodynamic lubrication (EHL) model with FGC, in which the elastic deformation and stress are computed using influence coefficients (ICs) and discrete convolution‐fast Fourier transform (DC‐FFT) algorithm. Comparisons are made with homogeneous EHL solutions and finite element analysis (FEA) to validate its accuracy. The study systematically explores how coating elastic modulus, coating thickness and substrate elastic modulus influence contact and lubrication behaviours. The developed model is expected to establish a theoretical framework for FGC material design, enhancing the performance of friction pairs.

Modeling of Multibody Lubrication Dynamics in Cylindrical Roller Bearings

As a typical multibody system, rolling bearing consists of several lubricated frictional pairs which transmit motions and forces among components. In this paper, a multibody lubrication dynamics (MLD) model of cylindrical roller bearings (CRBs) is developed by coupling the bearings dynamics sub-model and several lubrication sub-models of each frictional pairs in real-time. The lubrication sub-models are discretized by using finite difference and solved by numerical iteration methods of Gauss-Seidel with line relaxation and discrete convolution-fast Fourier transform (DC-FFT) technique. What’s more, a parallel algorithm is also designed to enhance the calculation efficiency. Its results are validated by the experimental results in literature, and further compared by the results from other dynamics models. It is revealed that the film formation changes not only friction but also geometrical constraints and pressure distribution, which may introduce additional load inside bearing and increase rolling friction under high-speed conditions. As the dynamic oil film stiffness and damper are took into account by MLD, the cage has a lower whirl frequency and forces. The curve-fitting formulas of EHL film thickness are overestimates the thickness value, and thus change the load distribution and friction between roller and raceway.

Impact of frictional heat and electric charge on the thermomechanical contact properties of piezoelectric quasicrystals: A DC-FFT algorithm

Piezoelectric quasicrystals (PEQCs) are quasicrystals with piezoelectricity. It combines the excellent properties of piezoelectric materials and quasicrystal materials, and is expected to be used as actuators in fields such as aerospace, automotive, and intelligent manufacturing. This article investigates the physical and mechanical properties of one-dimensional hexagonal piezoelectric quasicrystals (1D HPEQCs) coating under normal force and frictional heat and current density by using the discrete convolution-fast Fourier transform (DC-FFT) algorithm-based hybrid element method (HEM). The general solution of the model in the two-dimensional Fourier domain is obtained by combining the two-dimensional Fourier transform with strict operator theory and the Almansi’s theorem. The DC-FFT algorithm-based HEM is used to efficiently and accurately cope with the issue of transforming the fundamental solution in the Fourier domain into the spatial domain. Therefore, the numerical solutions and distribution of stress and displacement in the phonon and phason fields are obtained. The results indicate that current density has a significant impact on the phason field, and an increase in current density enhances the mechanical behavior of the phason displacement. The research results can provide theoretical guidance for 1D HPEQCs as intelligent coating materials.

On the time-dependent sliding contact behavior of three-phase polymer matrix smart composites

Abstract Three-phase smart composites consisting of magnetostrictive and piezoelectric reinforcements embedded with a polymer matrix can achieve specific multifunctional properties in response to external stimuli, which are well-suited for the application of sensors, actuators, and electronic devices. The materials exhibit complex behaviors characterized by electro-magneto-viscoelasticity coupling during the contact of these smart structures. This paper proposes a novel hybrid element method (HEM) for numerically analyzing the frictionless sliding contact problem stemming from the viscoelastic behavior and multiphase interactions of polymer matrix smart composites. The study aims to fully investigate the effects of material properties, sliding velocities, and action time on the contact behavior of materials via the integration of the conjugate gradient method (CGM) with the discrete convolution-fast Fourier transform (DC-FFT) algorithm. The analytical viscoelastic frequency response functions (FRFs) are derived by substituting elastic solutions with the time-dependent relaxation modulus. Numerical results show that three-phase polymer matrix smart composites exhibit lower contact pressure and higher surface electric/magnetic potential than three-phase magneto-electro-elastic composites. Sliding velocity and action time strongly influence the distribution of pressure/stress and electric/magnetic potential.

Dynamic responses of transversely isotropic and layered elastic media with imperfect interfaces under moving loads

Flexible pavement is widely used in engineering practice but is often subjected to the moving traffic loads, with imperfect contact behavior at the interfaces between adjacent layers. This study investigates transversely isotropic and layered elastic media with imperfect interfaces under moving vertical and horizontal loads using a semi-analytical method. The governing equation for moving loads is established within a Cartesian coordinate system and by virtue of the Galilean transformation, which is further decoupled into two ordinary differential equations in terms of the powerful Cartesian system of vector functions. General solutions for any layer are obtained, and the dual-variable position method is applied to derive the semi-analytical solutions for the layered pavement in the vector function domain. The lately introduced refined conversion algorithm, originally from the discrete convolution-fast Fourier transform (DC-FFT) algorithm, is applied to obtain the solution in the physical domain, which can efficiently remove the Gibbs effect near the source. The solutions are validated by comparison with existing solutions and numerical examples are presented to study the effect of interface modulus, moving load velocity, Young’s modulus of asphalt concrete and horizontal/vertical loading ratio on the surface dynamic response of the flexible pavement. Finally, the fatigue and rutting life of the pavement structures corresponding to different imperfect interface moduli are analyzed. The present solution provides practical guidance for the design of flexible pavement.

Numerical modeling of edge inhomogeneity in a macro cantilever beam

This study explores the inhomogeneity problem near the edge of a cantilever beam. Common in gears and turbine blades, inhomogeneities under local edge contact, coupled with macroscale bending effects, result in extreme and complex stress concentration. This model employs the Conjugate Gradient Method to combines the macroscopic beam theory and microscopic elasticity. The Eshelby inclusion method for simulating inhomogeneities. The Hetényi image method is used for address edge contacts in the beam. Enhanced by the Discrete Convolution fast Fourier Transform algorithm, the model’s accuracy and efficiency are validated with the Finite element method. Parametric analysis highlights the influence of distance between indenter and edge, stress components distribution and potential failure in cross-scale challenges.

A semi-analytical method of three-dimensional dual-phase-lagging heat conduction model

With the development of picosecond laser technology, the dual-phase-lagging (DPL) model is more suitable to describe the heat transfer problem with micro-scale effect and micro-time effect. In this paper, a three-dimensional dual-phase-lagging heat transfer model for picosecond laser processing is established. Then Fourier transform (FT) and Laplace transform (LT) are used to derive the frequency response functions (FRFs) of the problem in the frequency domain, and the inverse Laplace transform (ILT) and discrete-convolution fast Fourier transform (DC-FFT) algorithm are used to solve the temperature field. The calculation accuracy of this method is verified and the effect of phase lag constants on temperature field is studied. The results show that the semi-analytical method has a high accuracy for solving the dual-phase-lagging heat conduction model, and the phase lag of heat flux will affect the heating rate in the heating stage, and the phase lag of temperature gradient will delay the formation of temperature field. The effect of phase lag constants on temperature-time history is investigated and the related phenomena are explained by using the two-step heat transfer theory. This semi-analytical method provides a new approach to solve the problem of three-dimensional dual-phase-lagging heat transfer.

A semi-analytical solution for inhomogeneous material in the quarter space

A semi-analytical model is first presented that thoroughly examines stress distribution patterns in inhomogeneous materials within a quarter space: a scenario of great significance and complexity observed in components such as rail-wheels, roller bearings, and gears. The model utilizes the Hetényi Image Method and Eshelby’s Equivalent Inclusion Method, effectively handling the impact of inhomogeneities and satisfying the boundary conditions of the quarter space. To ensure efficient and accurate analysis, numerical techniques are incorporated, notably the Conjugate Gradient Method and the Discrete Convolution-Fast Fourier Transform algorithm. The model’s validity and robustness are demonstrated through comparisons with the Finite Element Method results, displaying excellent agreement. Parametric analyses reveal that distance, depth to the free surfaces, and shape of the inhomogeneities significantly influence the results. The introduction of an edge decay coefficient, Keg, serves as a practical tool for rapid estimation of edge effects in inhomogeneous materials, significantly contributing to precise predictions of material fatigue and failure.

A semi-analytical solution to incipient sliding contact on viscoelastic layer-elastic substrate with imperfectly bonded interface

This paper presents a semi-analytical solution to the incipient sliding contact between a rigid sphere and a viscoelastic layer-elastic substrate with spring-like interface. In this solution, Papkovich–Neuber potentials in frequency domain are applied to calculate the influence coefficients (ICs) based on the surface and interface conditions; the elastic constant in the elastic solution is replaced by the creep function to cover the time-dependent characteristics of the viscoelastic layer; discrete convolution-fast Fourier transform (DC-FFT) is used to obtain the displacements and stresses caused by the external load; conjugate gradient method (CGM) is employed to compute the pressure on the top surface of the viscoelastic layer. The validity of the solution is verified by a previous method and the finite element method (FEM), where the relaxation and creep functions of the viscoelastic layer are expressed as Prony series. Influences of variations in parameters on the surface pressure, normal displacement and von Mises stress are analyzed by a parametric study of the time, relaxation time, relaxation modulus, stiffness coefficients, coating thickness and friction coefficient. The results indicate that this solution can predict the contact behaviors of viscoelastic layered half-space and may serve as a numerical experiment in research of coating materials.

"THERMOELASTIC DISPLACEMENT AND TEMPERATURE RISE IN A HALF-SPACE DUE TO A STEADY-STATE HEAT FLUX "

Due to model complexity, classical contact mechanics theory assumes isothermal contact processes, involving bodies with uniform temperatures and no heat transmitted or generated through or near the contact interface. This paper addresses the problem of frictional heating in non-conforming or rough contacts by investigating the thermoelastic behaviour of asperities. The heat generated in a sliding contact by interfacial friction leads to thermoelastic distortion of the contact surface, further modifying contact parameters such as pressure, gap or temperature. The thermal expansion of the contacting bodies must therefore be accounted for when solving the contact problem. The thermoelastic displacement is computed with the aid of the half-space theory and of fundamental solutions for point sources of heat located at the free surface, derived in the literature of heat conduction in solids. The linearity of conduction equations encourages the use of superposition principle in the same way as for the elastic displacement. As the thermoelastic displacement is expressed mathematically as a convolution product, methods derived in contact mechanics for elastic displacement calculation are adapted to the heat conduction equations. The influence coefficients needed to efficiently compute the convolution products are derived, and the Discrete Convolution Fast Fourier Transform technique is applied to improve the algorithm computational efficiency. A similar method is then advanced for the temperature rise on the contact interface due to arbitrary heat input. The predictions of the newly advanced computer programs are tested against existing closed-form solutions for uniform circular or ring heat sources, and a good agreement is found.

Frictional contact of one-dimensional hexagonal quasicrystal coating considering thermal effects

In this paper, the contact behavior of a one-dimensional hexagonal quasicrystal coating fully bonded to a rigid substrate under the action of a spherical indenter rotating around its own axis is investigated. The frictional heat generation and heat transfer are considered. Using the general solutions of thermoelastic fields combined with thermal–mechanical boundary conditions followed by the double Fourier transforms, the frequency response functions of the basic solutions of the thermoelastic contact problem are obtained. The basic solutions in spatial domain are calculated by the discrete convolution-fast Fourier transform (DC-FFT) algorithm and the inverse fast Fourier transform (IFFT) algorithm. The results show that the asymmetry of responses of phonon normal stress, phason normal displacement and phason normal stress are mainly caused by friction behaviors. The increase of friction coefficient does not change the phonon normal displacement, but makes the phason normal displacement and the normal stresses of phonon and phason field opposite change trends on both sides of the contact center.

Steady state thermoelastic contact problem of one-dimensional hexagonal quasicrystals

This paper investigates the thermoelastic contact problem of a rigid adiabatic spherical indenter moving on a one-dimensional hexagonal quasicrystal half-space. The frequency response functions of thermoelastic field are derived by considering the thermo-mechanical boundary conditions and thermoelastic governing equations. By using the discrete convolution-fast Fourier transform (DC-FFT) algorithm, the thermoelastic field is calculated numerically, and the distributions of displacements, stresses and temperature rise are obtained. Meanwhile, the influence of different friction coefficients on the thermoelastic field is investigated. The results indicate that, due to the large tangential force and high temperature, the normal phason displacement contour lines tilt to a certain extent in the opposite direction of tangential load, in addition to the slightly larger response at the right end of the contact area. The results of this paper can serve as the theoretical basis for studying the contact problem of one-dimensional hexagonal quasicrystal.

FFT-ASSISTED ALGORITHMS FOR 3D LINE-CONTACT PROBLEMS

The line-contact is a particular type of contact with a contact length much greater than its width. Such contact scenarios can be treated in the frame of a two-dimensional plane-strain problem if the contacting surfaces can be considered nominally smooth. However, surface irregularities inherent to any manufacturing technique lead to a discontinuous contact area that differs from the one derived on the basis of the smooth profile assumption. It is therefore tantalizing to pursue the solution of a line-contact problem using an intrinsically three-dimensional (3D) model, which can only be numerical due to lack of general analytical solutions in contact mechanics. Considering the geometry of the line-contact, a major challenge in its numerical modelling is that the expected contact area is orders of magnitude larger in one direction compared to the other. This may lead to an unreasonably large number of grids in the contact length direction, which translates to a prohibitive computational burden. An alternative approach, employed in this paper, is to treat the line-contact as non-periodic in the contact width direction, but periodic in the contact length direction, with a period equal to the window required to capture and replicate the surface specific texture. This periodicity encourages the contact problem solution by spectral methods based on the fast Fourier transform (FFT) algorithm. Based on this idea, two methods are derived in this paper from the existing Discrete Convolution Fast Fourier Transform (DCFFT) technique, which was previously developed for purely non-periodic contact problems. A first algorithm variant employs a special padding technique for pressure, whereas a second one mimics the contribution of multiple pressure periods by summation of the influence coefficients over a domain a few times larger than the target domain. Both techniques are validated against the existing analytical Hertz solution for the line-contact and a good agreement is found. The advanced methods seem well adapted to the simulation of contact problems that can be approximated as periodic in one direction and non-periodic in the other.

Transient and steady-state viscoelastic contact responses of layer-substrate systems with interfacial imperfections

This paper reports the development of a novel semi-analytical model for solving the transient and steady-state contact responses of a rigid sphere sliding/rolling on a viscoelastic layer-elastic substrate system. The displacement transmissions at the layer-substrate interface are affected by spring-like or dislocation-like defects. The analytical transient and steady-state viscoelastic frequency response functions (FRFs) are derived from the elastic solutions with imperfect interfaces. Instead of using the integration form of the creep function, viscoelastic modulus Ε(ω) is directly incorporated into the viscoelastic FRFs by a frequency-velocity transform that links the time-related frequency, ω, and sliding velocity, V, with the space-related frequency number, m, i.e. ω=−mV. The solutions are so formulated that fast numerical techniques, such as the conjugate gradient method (CGM) and the discrete convolution-fast Fourier transform (DC-FFT) algorithm, can be incorporated for computation efficiency. The developed model is employed to investigate the effects of layer thickness, modulus, sliding velocity, and the degree of interface imperfection on the viscoelastic contact response of the material system, including pressure distributions, displacements, viscoelastic dissipation, and subsurface stresses.

Couple stress-based 3D contact of elastic films

This study reports a three-dimensional (3D) size-dependent contact model of a thin elastic film indented by an elastic sphere, based on a linear couple stress-based semi-analytical method (SAM), where the thin film is perfectly bonded to a rigid substrate. The couple stress-based frequency response functions (FRFs) of the film are analytically derived, and the corresponding continuous and discrete influence coefficients (ICs) are subsequently obtained. The contact pressure is numerically determined based on the conjugate gradient method (CGM), while the elastic displacements and stresses of the thin film and indenter are calculated by means of the discrete convolution-fast Fourier transform (DC-FFT) algorithm. The combined effects of the characteristic material length and film thickness on contact behaviors are investigated for selected material elastic coefficients.

A SAM-FFT based model for 3D steady-state elastodynamic frictional contacts

This paper reported a semi-analytical method (SAM)-fast Fourier transform (FFT) based model for three-dimensional (3D) steady-state elastodynamic frictional contact of an elastic ellipsoid sliding on an elastic half-space with a constant sliding velocity. The frequency response functions (FRFs) and their conversion into influence coefficients (ICs) for displacements and stresses in an elastic half-space are analytically derived pertaining to generalized normal and tangential forces. Fast numerical techniques used are based on the conjugate gradient method (CGM) for obtaining unknown pressure distribution in the contact interface, and the discrete convolution-fast Fourier transform (DC-FFT) algorithm for calculating displacements and stresses. The proposed SAM-FFT based model is employed to investigate the effects of friction, sliding velocity, and Young's modulus on contact pressure, surface deformation and sub-surface von Mises stress. A transition map, supported by appropriate limits of friction coefficient and sliding velocity, is constructed to determine whether the location of maximum von Mises stress to appear beneath the contact surface or in the contact surface. It deserves mentioning that the elastodynamic effect becomes more profound if the sliding velocity is higher than 0.4 times of shear wave speed, which corresponds to a sliding velocity of 1300 m/s for steel materials (shear wave speed ∼3250 m/s), or 60 m/s for a soil foundation (shear wave speed ∼150 m/s).

A Thermal Elastohydrodynamic Lubrication Model for Crowned Rollers and Its Application on Apex Seal–Housing Interfaces

This paper presents a thermal elastohydrodynamic lubrication (EHL) model for analyzing crowned roller lubrication performances under the influence of frictional heating. In this thermal EHL model, the Reynolds equation is solved to obtain the film thickness and pressure results while the energy equation and temperature integration equation are evaluated for the temperature rise in the lubricant and at the surfaces. The discrete convolution fast Fourier transform (DC-FFT) method is utilized to calculate the influence coefficients for both the elastic deformation and the temperature integration equations. The influences of the slide-to-roll ratio (SRR), load, crowning radius, and roller length on the roller lubrication and temperature rise are investigated. The results indicate that the thermal effect becomes significant for the cases with high SRRs or heavy loads. The proposed thermal EHL model is used to study the thermal-tribology behavior of an apex seal–housing interface in a rotary engine, and to assist the design of the apex seal crown geometry. A simplified crown design equation is obtained from the analysis results, validated through comparison with the optimal results calculated using the current crowned-roller thermo-EHL (TEHL) model.

Numerical simulation of elastic bilayered contact. Part I – Contact parameters

Protective coatings are increasingly utilized to provide low friction, high wear resistance and long fatigue life to many tribological applications including cutting tools, magnetic data storage or engine components. The coating design relies heavily on the knowledge of contact stresses generated in rough-surface contacts, thus requiring improved contact models and efficient numerical methods. Spectral analysis, based on the fast Fourier transform (FFT) and on the convolution theorem, has played an important role in assessing the elastic response of a homogenous half-space, as the corresponding Green’s functions are readily available in the time/space domain. In the case of a bilayered medium, the elastic response is only known in the frequency domain as the frequency response functions, and derivation of the Green’s functions can be challenging. This paper explores the possibility of using spectral methods in the analysis of contact behaviour of layered elastic solids. The starting point is the Discrete Convolution Fast Fourier Transform (DCFFT) technique, which eliminates the periodicity error associated with application of FFT to non-periodic problems. The influence coefficients required by DCFFT are derived from the frequency response functions. The numerical predictions agree well with results from the literature of layered elastic solids under normal contact load.

A mixed elastohydrodynamic lubrication model based on virtual rough surface for studying the tribological effect of asperities

PurposeThe main purpose of this paper is to present the effort on developing a mixed elastohydrodynamic lubrication (EHL) model to study the tribological effect of asperities on rough surface.Design/methodology/approachThe model, with the use of the average flow Reynolds equation and the K-E elasto-plastic contact model, allows predictions of hydrodynamic pressure and contact pressure on the virtual rough surface, respectively. Then, the substrate elastic deformation is calculated by discrete convolution fast-Fourier transform (DC-FFT) method to modify the film thickness recursively. Afterwards, corresponding ball-on-disk tests are conducted and the validity of the model demonstrated. Moreover, the effects of asperity features, such as roughness, curvature radius and asperity pattern factor, on the tribological properties of EHL, are also discussed though plotting corresponding Stribeck curves and film thickness shapes.FindingsIt is demonstrated that the current model predicts very close data compared with corresponding experimental results. And it has the advantage of high accuracy comparing with other typical models. Furthermore, smaller roughness, bigger asperity radius and transverse rough surface pattern are found to have lower friction coefficients in mixed EHL models.Originality/valueThis paper contributes toward developing a mixed EHL model to investigate the effect of surface roughness, which may be helpful to better understand partial EHL.

An efficient model for the frictional contact between two multiferroic bodies

This paper presents a semi-analytical model (SAM) for three-dimensional frictional magnetoelectroelastic (MEE) contact of two multiferroic bodies, together with a set of effective solution methods. The frequency response functions (FRFs) for the MEE fields in a multiferroic half-space are analytically derived with respect to a unit concentrated normal force, a unit concentrated tangential force, a unit electric charge, and/or a unit magnetic charge, which are then converted into the results of continuous Fourier transforms of the influence coefficients (ICs), followed by the discrete Fourier transforms with a proper aliasing treatment. The conjugate gradient method (CGM) is used to obtain the unknown distributed pressure. Furthermore, the discrete convolution-fast Fourier transform (DC-FFT) algorithm is implemented to calculate the in-plane electric/magnetic potentials and subsurface stresses. The model is implemented to analyze the frictional sliding contact between a half-space and a sphere, and to study the coupled effects of surface electric/magnetic charges and friction on contact behaviors, including pressure, stresses, and electric/magnetic potentials. A sensitivity analysis is also conducted to evaluate the influences of friction and material properties on the contact-induced multifield coupling behaviors. A number of case studies are committed, and the results indicate that electric/magnetic charge densities and the friction coefficient strongly influence the contact pressure, stress, and electric potential.